Student Task:

Assumed that students have a working knowledge of the program geometer's sketchpad, specifically they can draw circles, construct lines, label points,and create math equations related to the diagram.

Have completed investigations on interior angles and circles.

Terms perpendicular, intersection, secant, tangent, nolocus, radius, diameter, point, cyclic, concyclic, Quadralaterial.

Student investigation — Tangents and Circles

Using Geometer's Sketchpad you will be constructing circles and points outside a circle. You will use these objects to investigate properties between the two. (note: make your circles small enough to allow room to see the investigations )

Topic one: What is tangent?

Construction

  1. Construct a circle: label the center
  2. Construct a point outside the circle, and label
  3. Draw a line segment from labeled point through the circle creating a secant line
  4. Mark the intersection points
  5. Animate the secant line until the two points come together, Mark this point
  6. Draw a line from the centre to the point labeled in step 5
  7. Measure the angle between centre point and the single point found in step 5

Investigation

  1. when a line crosses a circle at two points it is called _____________________
  2. when the two points come together to form a single point what is the point called ______________
  3. Tangent means______________________________________________________________________
  4. How many tangent points are their between a point and a circle ___________
  5. What is the angle between a line passing through the centre to the point of tangency and a line form an exterior point and the point of tangency ___________
  6. State the tangent theorem and its converse
  7. Test your theory with using a second point
  8. If you where given a line that appeared to touch a circle at one point how would you guarantee it was tangent

Print out your diagram showing the points of tangency the radius lines and the exterior point

Tangent Chord Theorem

In this lab you will draw a circle with a line tangent to the circle. You will then place a chord inside the circle. Your objective is to prove that the angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord

Construction

  1. Construct a circle label centre O
  2. Locate a tangent line make sure to give a proof that it is tangent label tangent point P
  3. Locate a point B on the circle
  4. Draw a chord from tangent point P to B
  5. Locate a point A on the opposite side of the circle Draw a line form point P to Point A

Measurements

  1. Measure the angle between chord PB and tangent line
  2. Measure the angle between chord AP and BP
  3. Measure the angle between the tangent line and the radius coming from the line to the centre point

Investigation

Using point B move around the circle, observe the measurement of the angle

Conclusions

Write the Tangent Chord theorem?

What is the relation of an angle at the chord to the centre angle?

Try to solve these problems. You can use sketchpad make sure you print out your solutions

If the radius of the smaller circle is 8cm find the radius of the larger. Assume the two circles touch at one point and have two common tangents

2.

Find all interior angles and both angles between tangent and chord

Cyclic or Concyclic Shapes

In this lab you will be trying to draw cyclic quadrilateral like the one below. You will then find the relationship between interior angles A cyclic shape is one in which all vertices are located on the circumference of a single common circle

Cyclic Quadrilateral Non Cyclic Quadrilateral

Construction

  1. Draw a circle. define and label 4 points on the circle
  2. Connect the points to form a quadrilateral

Measurements

  1. measure all interior angles
  2. measure all exterior angles

Investigation

  1. What is the sum the interior angles?
  2. What is the relation ship between opposite angles?
  3. What is the relation ship between the interior and exterior angles?
  4. Change one of the point’s placements on circle. Does the relationships found in Question 2,3 change?
  5. Write a theorem for cyclic quadrilaterals?
  6. How would you prove four points are concyclic?

Exsemplar student work Topic 1

Investigation answers

  1. secant lines
  2. tangent point
  3. to touch at one point
  4. two
  5. 90 degrees
  6. A line that is tangent to a circle will intersect at 90 degrees to a radius line from the point of tangency

8. check the angle between the line and the radius if it is a right angle then it is tangent

Student project 2

Observation that the angles are the same

The angle between a tangent and a chord is equal to the inscribed angle on the opposite side of the chord

Student Project 3

(No image available)

  1. 360 degrees
  2. the are supplementary add up to 180
  3. the are supplementary
  4. no
  5. the opposite angles are supplementary on a cyclic quadrilateral
  6. measure interior angles see if they are 360 degrees